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On stickiness of multiscale randomly rough surfaces
Article in The Journal of Adhesion · November 2019
DOI: 10.1080/00218464.2019.1685384
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On stickiness of multiscale randomly rough surfaces
G. Violano,1 L. Afferrante,1 A. Papangelo,1, 2 and M. Ciavarella1, 2
1Department of Mechanics, Mathematics and Management,
Polytechnic University of Bari, Via E. Orabona, 4, 70125, Bari, Italy
2Hamburg University of Technology,
Department of Mechanical Engineering,
Am Schwarzenberg-Campus 1, 21073 Hamburg, Germany
Abstract
A new stickiness criterion for solids having random fractal roughness is derived using Persson’s
theory with DMT-type adhesion. As expected, we find stickiness, i.e., the possibility to sustain
macroscopic tensile pressures or else non-zero contact area without load, is not affected by the
truncation of the PSD spectrum of roughness at short wavelengths and can persist up to roughness
amplitudes orders of magnitude larger than the range of attractive forces. With typical nanometer
values of the latter, the criterion gives justification to the well-known empirical Dalhquist criterion
for stickiness that demands adhesives to have elastic modulus lower than about 1 MPa.
Keywords: stickiness criterion, adhesion, Dalhquist criterion, DMT model
1
I. INTRODUCTION
Contact mechanics with roughness has made tremendous progress in recent years (for
two recent reviews, see Ref. [1, 2]), and adhesion has become increasingly relevant with
the interest on soft materials [3], nano-systems [4, 5] and the analysis of bio-attachments
[6, 7]. Contact between solids occurs via large van der Waals forces, usually represented,
for example, by the well known Lennard-Jones force-separation law. These forces give rise
to a theoretical strength much higher than the typical values to break bulk materials apart.
Hence, the ”adhesion paradox” [8] states that all objects in the Universe should stick to
each other. This does not happen due to inevitable surface roughness at the interface, and
Nature has developed different strategies to achieve stickiness, including contact splitting
and hierarchical structures [9–11]. At macroscale and for nominally flat bulk solids, it
appears that the only solution to maintain stickiness is to reduce the elastic modulus. This
is well known in the world of Pressure-Sensitive Adhesives (PSA), soft polymers showing
instantaneous adhesion on most surfaces, upon application of just a light pressure [12, 13].
Dahlquist [14, 15] proposed that to achieve a universal stickiness, the elastic Young modulus
should be smaller than about 1 MPa (at 1 Hz, as adhesive are strongly viscoelastic their
modulus depends on frequency). This criterion has no scientific validation, but appears to
be largely used in the world of adhesives.
There have been various attempts to study the problem of elastic contact with roughness
and adhesion. Fuller and Tabor (FT, [16]) used the Greenwood and Williamson [17] concept
of describing a rough surface with a statistical distribution of identical asperities of radius
R, together with JKR theory for the sphere contact [18]. FT found that adhesion was easily
destroyed with root mean square (RMS) amplitude of roughness hrms of a few micrometers in
spherical rubber bodies against rough hard Perspex surfaces. Their theory depends only on a
single dimensionless parameter θFT = h3/2rms∆γ/
(R1/2E∗
)where E∗ is the plane strain elastic
modulus, ∆γ is the interface energy. The choice of R seems critical in view of its sensitivity
to ”resolution” or ”magnification” [19], i.e., to the shortest wavelength in the roughness
spectrum. In the ”fractal limit”, i.e., for an infinite number of scales, R → 0, there would
be no stickiness for any surface, in the sense that the solution would be identical to that
without adhesion, irrespective of the geometrical characteristics, like fractal dimension, or
RMS amplitude of roughness. Hence, FT apparent good correlation with the theory despite
2
the many limitations (see [20]), may have been due to a fortuitous choice of R at a relatively
coarse scale where measurements were made at that time.
The JKR theory is inappropriate when contact spots become very small, and another
theory is more promising in this case, which takes the name of DMT, stemming for the
original case of the sphere [21]. Joe, Scaraggi and Barber [22] showed that the JKR approach
becomes questionable for contact problems involving fine-scale roughness; in such case a
solution based on the full Lennard-Jones traction law is more appropriate. Moreover, Violano
and Afferrante [23, 24] showed that DMT-type models are accurate in predicting the effective
interfacial binding energy, when compared with calculations including the ”exact” Lennard-
Jones law. DMT makes possible to solve contact problems with adhesion using results from
the adhesionless problem, by assuming that the adhesive stresses do not alter the pressure
in the contact area (which therefore remains purely under compression, and remains defined
in the same way as ”repulsive”) nor the gaps outside the contact. The external pressure is
therefore the difference of the repulsive and an adhesive pressure pext = prep − pad.
Pastewka & Robbins ([25]) presented a criterion for adhesion between randomly rough
surfaces after interpreting simulations of adhesive rough contact with fractal roughness,
which were obtained on spectra of roughness 3 orders of magnitude in wavelengths (from
nano to micrometer scale). They determined the surface ”stickiness” based on the slope
of the curve repulsive area vs external load (a definition which we shall also adopt here)
elaborating a simplified DMT-like model that uses only the asymptotic expression for gaps
at the edge of the repulsive contact regions, hence defining an ‘adhesive boundary layer’ that
surrounds the ‘repulsive’ contact zone. They obtained a criterion for stickiness that depends
mainly on small scale features of the rough surface, i.e., on local slopes and curvature, and
shows to be in agreement with their numerical calculations.
Recent theories by Ciavarella [26] with its BAM model (Bearing Area Model) and Joe,
Thouless and Barber (JTB, [27]), estimate the pressure at pull-off between surfaces does not
depend much on local slopes and curvature, and while the former theory is a very simple
model, the latter involves a full recursive solution using the Lennard-Jones potential. Similar
results are found in Ref. [28], where the effect of surface topograghy on the pull-off force is
investigated in the framework of a DMT theory. Also, Refs. [29, 30] using JKR framework,
suggest that the contact area reaches a constant value as the magnification is increased and
full contact occurs at the short length-scale structures of the surface.
3
In this paper, we use the Persson and Scaraggi theory (PS theory, [31]) based on the DMT
assumption, with some refinements [32], to derive a ”stickiness” criterion for very broad
spectra, typical of real surfaces which can be expected to have features from millimeter (or
more) to nanometer scale.
II. METHODS
A. The rough contact model
In DMT theories, the adhesive pressure is computed by convolution of the elementary
tension-separation law σad (g) with the distribution of gaps P (g), obtained by a standard
non-adhesive contact solution. In general, the Lennard-Jones force-separation law is usually
represented as
σad(g) =8∆γ
3ε
[ε3
g3− ε9
g9
](1)
where g is the local gap, ∆γ =∫∞εσ(g)dg is the interface energy or, by definition, the work
done per unit area of interface in separating two bodies from the equilibrium position g = ε,
at which σad = 0. The maximum tensile traction happens at a separation g = 31/6ε and
is σth = 16∆γ/(9√
3)ε. A possible simplification, which will be adopted in the following
derivations, is to use a constant force-law [33]. Considering gaps from the equilibrium point
namely u = g − ε, imposing the same interface energy ∆γ,
σad (u) = σ0, u ≤ ε
σad (u) = 0, u > ε(2)
where σ0 = 9√
3σth/16 ' σth and ε is the same range of attraction, so obviously ∆γ = σ0ε.
Notice that if E∗ = E/ (1− ν2) is the plane strain elastic modulus (E is the Young’s
modulus and ν is the Poisson’s ratio), then la = ∆γ/E∗ defines a characteristic adhesion
length which can be used to quantify the relative strength of adhesion. The value of la/ε
is of the order of the fractional change in bond length needed to change the elastic energy
by the binding energy, and la/ε � 1. For example, the typical Lennard-Jones description
of an interface has la ' 0.05ε. The theoretical strength in this case, σ0 = laE∗/ε = 0.05E∗
represents a very high value.
4
In DMT convolution of the elementary tension-separation law σad (u) with the distribution
of gaps P (u), for the Maugis potential, simplifies therefore to
pad = σ0
∫ ε
0
duP (u) = σ0AadAnom
(3)
where Aad is the ”adhesive” contact area, i.e. the region where tensile stresses are applied,
and Anom is the ”nominal” or ”apparent” contact area. An elaborate expression for P (u)
(for the purely repulsive problem, i.e. in the absence of any adhesion) is obtained in Persson’s
theory (see Appendix - A eq. (29) and [32, 34]).
FIG. 1: (A) Distribution of gaps P (u) (black, blue and red line respectively for ζ = 10,100, 1000) as obtained by Persson’s theory. An asymptotic fit P (u) ∼ u−1/3 is also plottedas guide to the eye (dashed black line). Results are obtained for a self-affine fractal surfacewith a pure power law PSD, fractal dimension D = 2.2 (or Hurst exponent H = 0.8), andinterfacial mean separation u/hrms = 2. (B) Dependence of the P (u) on the magnificationζ. Notice a convergence of P (u) with increasing magnifications for various values of thelocal gap u/hrms. Results are obtained for a self-affine fractal surface with a pure powerlaw PSD, fractal dimension D = 2.2 (or Hurst exponent H = 0.8), and interfacial mean
separation u/hrms = 1.37.
Fig. 1A-B shows that there is a convergence in the distribution of P (u) for increasing
magnification ζ = q1/q0, being q1 and q0 respectively the high and low wavenumber cut-off
of the surface Power Spectral Density (PSD). Furthermore there is an asymptotic scaling at
low separations P (u) ∼ u−1/3. Results in Fig. 1A are given for pure power law PSD, fractal
5
dimension D = 2.2 (i.e. Hurst exponent H = 3 −D = 0.8) and for a mean gap u equal to
twice the RMS amplitude of roughness, u/hrms = 2. We define a non-dimensional pressure
prep = prep/ (E∗q0hrms) and we remark that, for typical real surfaces H & 0.6, in the limit of
relatively large ζ and small pressures, Persson’s theory reduces to prep ' 34
exp (−2u/hrms)
[35, 36]. As we are essentially interested in the region of the area-load relationship near the
axes origin, we disregard u/hrms < 1, and also u/hrms > 3 where we are likely to have finite
effects due to poor statistics of the Gaussian surfaces and very few asperities in contact.
This corresponds therefore to the range prep = 10−3 − 10−1.
Furthermore, within our DMT hypothesis, the range of attractive forces of interest is
ε << hrms, where we can assume that the main contribution to the gaps and hence to
adhesion comes from the asymptotic value of P (u) at low u, namely from the regions close
to the contact boundaries. We can obtain more understanding of this asymptotic form of
P (u) from standard contact mechanics theory, and from the asymptotic part of the original
Persson’s theory [37], whereas we shall use the full Persson and Scaraggi theory only for the
actual calculation of the prefactors. Notice that these asymptotic derivations are similar
to what suggested by Pastewka and Robbins [25], except that we shall not make their
assumptions, which lead to the different final results.
B. Asymptotic results
Here we derive an asymptotic theory for the attractive area Aad. It is well known that
the relationship between (repulsive) contact area ratio Arep/Anom and mean pressure prep
ArepAnom
' κrepprep
E∗√
2m2
(4)
is linear for Arep/Anom up to almost 20% − 30% [37]. In the original Persson’s theory,
κrep = 2√
2√π
= 1.6, and we can also write V = 12E∗2m2 as the variance of full contact
pressures. Here, m2 is the mean square profile slope along any direction (for a isotropic
surface). The distribution of pressures P (p) near the boundaries of contact (on the contact
side) is at low p [38]
P (p) ' p prepV
√2
πV(5)
Suppose the perimeter of the actual contact area [not necessarily simply-connected] is Π.
6
We define position on Π by a curvilinear coordinate s. In view of the asymptotic behaviour
at the edge of the contact area [39], we must have at every point on Π, pressure p and gap
u as
p (x) = B (s)x1/2; u (x) = C (s)x3/2; (6)
where x is a coordinate perpendicular to the boundary and C (s) = βd (s)−1/2 where β is a
geometrical prefactor of order 1, B (s) = 3E∗βd (s)−1/2 /4 and d (s) is a local characteristic
length scale.
It is clear that, as load is increased, existing contacts grow larger and new contacts form,
resulting in a very irregular shape. Already Greenwood and Williamson [17] in their simple
asperity theory suggested that the average radius of contact should remain constant with
load, as a result of competition between growing contacts and new contacts forming, and
this is correctly captured despite the strong approximations in the asperity model. In Ref.
[40] it was shown, again with a simple asperity model, but with an exponential distribution
of asperity heights, d seems indeed completely independent on load.
Hence, we shall leave the quantity d (s) to vary arbitrarily along the perimeter. For a
segment ds of Π, there exists a region dxds = 2pdpds
B(s)2in which the pressure is in the range (p,
p+dp). It follows that the total area with the pressure is∫S
2pdpds
B(s)2, and hence the probability
of an arbitrary point being in this range is P (p) dp and the PDF is
P (p) =2p
Anom
∫Π
ds
B (s)2 =2pΠ
AnomIp (7)
where Ip =∫ 1
0ds/B (s)2 = [4/ (3βE∗)]2 〈d〉, s = s/Π is a normalized coordinate along the
perimeter, and 〈d〉 means the mean value of d. A similar argument with the gap expression
yields
P (u) =1
Anom
(4
9u
)1/3
ΠIu (8)
where Iu = β−2/3∫ 1
0d (s)1/3 ds = β−2/3
⟨d1/3
⟩. Eliminating the perimeter from eqs. (7, 8)
and using eq. (5), we obtain Π = pAnom
2Ip
√2
πV 3/2 and hence
P (u) =
(4
9u
)1/3Iu2Ip
√2
πV 3prep (9)
7
Upon integration of the distribution of gaps P (u), the attractive area can be given as
AadAnom
=3
2aV prep
(ε
hrms
)2/3
(10)
being the coefficient aV
aV =3
2
(4
9
)1/39
32β8/3
⟨d1/3
⟩〈d〉
E∗3√
2
πV 3q0h
5/3rms (11)
Notice, in eq. (10), Aad/Anom is proportional to the external mean pressure if⟨d1/3
⟩/ 〈d〉
does not depend on pressure.
III. RESULTS
We have obtained an expression for aV in eq. (10), but this was mainly for qualitative
purposes because of the problematic term⟨d1/3
⟩/ 〈d〉. Hence, we fit the global Aad/Anom
vs. (ε/hrms)2/3 curves obtained from numerical results of the adhesionless contact problem
with the full Persson-Scaraggi’s theory. The use of the Persson-Scaraggi’s theory allows us
to reach broad band of roughness, ζ ' 105. Fig. 2 shows that in terms of actual adhesive
area, the convergence with ζ is very rapid (Fig. 2A) and is not modified by the load (Fig.
2B). Accordingly, interpreting the results in terms of the prefactor aV , the solution rapidly
converges with magnification (Fig. 2C), and weakly depends on pressure at the smaller
values of prep (Fig. 2D) or indeed on fractal dimension in the range D = 2.1 − 2.3, which
is the most interesting range [41]. We also show Pastewka and Robbins’ prediction for the
prefactor aV , which corresponds to aV,PR ∼ ζ1/3 for D = 2.2 (see Appendix B) and hence
may result in large error for large magnifications.
Starting again from eq. (10), we find that both repulsive mean pressure and adhesive
mean pressure are proportional to the repulsive contact area, which can therefore be grouped
as
pextE∗
=prepE∗− σ0
E∗AadAnom
=ArepAnom
√2m2
2
[1− la
ε
3
2
aVq0hrms
(ε
hrms
)2/3] (12)
8
FIG. 2: (A) Adhesive area Aad/Anom estimated by Persson’s theory (withζ = [10, 100, 500, 1000, 5000, 10000, 20000, 50000] , H = 0.8) . (B) Adhesive area Aad/Anom
estimated by Persson’s theory (withprep = [6.5 ∗ 10−3, 0.03, 0.07, 0.13, 0.20, 0.26, 0.33, 0.39, 0.46, 0.52] and ζ = 1000). (C)
Estimates of the prefactor aV from Persson’s theory for H = [0.7− 0.8− 0.9] vs Pastewkaand Robbins’ estimate for H = 0.8 (black dashed line), [aV ]PR = 0.252ζ1/3 as obtained in
Appendix - B as a function of magnification ζ for prep = 0.065. (D) Estimates of theprefactor aV for ζ = 1000 from Persson’s theory as a function of for prep.
9
where we used the identity la/ε = σ0/E∗ and the actual value of the factor κrep = 2 [42, 43]
instead of the factor of the original Persson’s theory.
Hence for the ”slope” κ we write
1
κ=
1
κrep− 1
κad=
1
2− 3aV
4q0hrms
laε
(ε
hrms
)2/3
(13)
and then stickiness is obtained when 1/κ < 0 leading to the suggested criterion
ε
la
(hrmsε
)2/3
<3
2
aVq0hrms
(14)
In particular, neglecting the weak dependence on fractal dimension (see Fig. 2C), considering
we are interested in the asymptotic regime of low pressure (see Fig. 2D) and in the limit of
high magnification (see Fig. 2C), we can take aV ' 3 and rewrite the criterion given in eq.
(14) as
hrmsε
<
(9
4
la/ε
εq0
)3/5
(15)
As it can be seen eq. (15) depends only on RMS amplitude of roughness hrms and the largest
wavelength in the roughness spectrum q0.
The obtained criterion depends only on quantitative macroscopic entities, rather than
local, small-wavelength dependent ones. The small wavevector cutoff of roughness q0, could
in principle take arbitrarily low values for large surfaces, provided that for a given hrms/ε
the PSD components should be lowered over all the wavenumber q, which would result in
increasingly loose boundary of stickiness. For example, for a flat surface of mm size, we have
εq0 ∼ 10−6 and our criterion gives hrms/ε . 1000, so macroscopic stickiness could persist
even for roughness three orders of magnitude larger than the range of attraction, i.e. of the
order of nearly one micron. Of course these extrapolations will have some limitation on the
concept of the ideally flat surface with a pure power law PSD of roughness.
Sticky surfaces in Pastewka and Robbins’s simulations [25] have la/ε = 0.05, εq0 =
2π/4096, hence the right hand side of eq. (15) gives(
94la/εεq0
)3/5
' 13, while from [44, 45]
it is possible to estimate hrms/ε ' [5 − 10] which means our stickiness criterion (eq. (15))
was also satisfied for their “sticky surfaces”. Nevertheless the criterion we propose does not
depend on small scale features of the rough surface and coincides with that of Pastewka &
10
Robbins [25] only for ζ ≈ 103 (see Fig. 2C and for a more detailed comparison see Appendix
- B).
Obviously, our criterion may suffer from the limitations and assumptions of Persson and
Scaraggi’s theory, in particular those of the DMT theory. Indeed, from JTB’s results [27], we
know that complex instabilities and patterns form at very low RMS amplitude of roughness,
and in particular that any DMT type of analysis, including the one presented here, can be
expected to hold only for approximately
hrmsε
>4
75
la/ε
εq0
(16)
It is also likely, from JTB predictions [27], that stickiness is lower if this condition is violated
and a more refined analysis is needed, which is outside the possibilities of both JTB and our
model. Fig. 3 shows the stickiness map obtained with the present criterion. The shaded
region in Fig. 3A identifies the couple ( εq0la/ε
,hrms
ε) which would give stickiness. Underneath
the dashed line (from eq. (16)) pattern formation is expected, and our DMT model may
not be accurate. In Fig. 3B the slope angle α = arctan (κ) is plotted as a function of hrms/ε
for varying εq0la/ε
.
IV. DISCUSSION
Considering our result, it is clear that to improve stickiness, for a given range of attractive
forces ε, we need to make q0 as small as possible for the given hrms. For a power law PSD
C (q) = C0q−2(H+1), we have hrms '
√πC0/Hq
−H0 . It may be useful to rewrite the criterion
(15) in terms of the PSD multiplier C0 (as usual, for H = 0.8)
C0 <0.8
πε4/5q
2/50
(9
4
laε
)6/5
(17)
It appears clear that we need as small roughness as possible, for a given q0 which is presum-
ably dictated by size of the specimen up to some extent, or by the process from which the
surface originates. Also, we need to have la/ε as high as possible, and this means obviously
high ∆γ and low E∗ (being la = ∆γ/E∗).
Given ∆γ is in practice strongly reduced by contaminants and various other effects to
11
FIG. 3: (A) Adhesion map for multiscale rough surfaces according to the present criterion(15). In the plane εq0
la/εvs hrms
εthe sticky region is shaded. According to JTB analysis in the
region below the dashed black line pattern formation is expected with possible reduction ofstickiness. (B) The slope angle α = arctan (κ) is plotted as a function of hrms/ε for varying
εq0la/ε
= [1, 3/4, 1/2, 1/4, 1/10] and aV = 3. The arrow denotes increasing values of εq0la/ε
.
values of the order ∼ 50 mJ/m2, the only reliable way to have high stickiness is to have very
soft materials.
Specifically, we shall make some quantitative estimates based on real surfaces. As re-
ported by Persson [41], most polished steel surfaces for example, when measured on L ∼ 0.1
mm, show hrms ∼ 1 µm. This means q0hrms ∼ 0.1. This incidentally satisfies JTB condition
for ”DMT”-behaviour (eq. (16)). Contrary to the recent emphasys on measuring entire PSD
of surfaces, it seems therefore that for stickiness, the most important factors are well defined
macroscopic quantities, which are easy to measure. For a wide range of surfaces (asphalt,
sandblasted PMMA, polished steel, tape, glass) reported in [41], see Fig. 4, C0 is at most
5× 105 m0.4. Then, assuming ε = 0.2 nm and ∆γ = 50 mJ/m2 as a typical value, even with
the uncertainty in the choice of q0, in the range of q0 ∼ 103 [m−1], our criterion predicts
E∗ < 2.4 MPa. This is for asphalt which clearly is one of the most rough surface we can
consider, and indeed outside the normal application of PSA. Our result is therefore entirely
compatible with the empirical criterion by Carl Dahlquist [14] from 3M which suggests to
make tapes only with low modulus materials E∗ < 1 MPa, whose generality was so far still
12
scientifically unexplained.
FIG. 4: Experimentally measured PSD of typical real surfaces as taken from Ref. [41].Notice they can be grouped in a fairly narrow band.
Recently, after a recent paper by Dalvi et al. [48], which has demonstrated that the simple
energy idea of Persson and Tosatti [49] works reasonably well, Ciavarella [50] compared the
present DMT criterion with other ideas, namely one coming from an energy balance due to
Persson and Tosatti [49], where the reduction in apparent work of adhesion equals the energy
required to achieve conformal contact, and another criterion derived from BAM (Bearing
Area Model) of Ref. [26]. It was found that the three criteria give very close results but the
present criterion contains a slightly different qualitative dependence on material properties
than others, since we can write it in the form
hrms < ε−1/5 (0.36laλL)3/5 (18)
being λL = 2π/q0. The above formula shows the product laλL, is raised to the power 3/5,
and there is a weak apparent dependence on the range of attractive forces ε.
13
1 100 104 1061
10
100
1000
104
105
la
Ε
ΛL
Ε
Hhrm
s�ΕL t
hre
sh
NOT STICKY
STICKY
Ζ=103,104,105,106,107
FIG. 5: A comparison of the derived stickiness criterion (red line), together withPersson-Tosatti (black line), BAM (blue solid line) and that of Pastewka and Robbins [25]
(black dashed lines) which correlates well with ours only for ζ < 1000.
We can summarize the three criteria for H = 0.8 in the form
hrmsε
<
√0.24
laε
λLε
; Persson-Tosatti (19)
hrmsε
<
√0.6
laε
λLε
; BAM (20)
hrmsε
<
(0.358
laε
λLε
)3/5
; Violano et al. (21)
and a comparison is shown in Fig. 5, where Persson-Tosatti is reported in black solid line,
BAM as blue solid line, and Violano et al. as red solid line.
Clearly, the criterion obtained in the present paper deviates from the other two at high
la/ε, which corresponds to low elastic modulus, where indeed one expects that the DMT
assumptions may be violated. Notice that the PR criterion for H = 0.8 writes as
hrmsε
<
(0.06
laε
λLε
)3/5
ζ1/5 (22)
It is reported in Fig. 5 and shows similar behaviour from ours but only for low magnifi-
14
cations
V. CONCLUSIONS
We have defined a new stickiness criterion, whose main factors are the low wavevector
cutoff of roughness, q0, the RMS amplitude of roughness hrms and the ratio between the work
of adhesion and the plain strain Young modulus. We find that, in principle, it is possible to
have effective stickiness even with quite large RMS amplitudes, orders of magnitude larger
than the range of attractive forces. For robust adhesion with different possible levels of
roughness, the main characteristic affecting stickiness is the elastic modulus, in qualitative
and quantitative agreement with Dahlquist criterion, well-known in the world of pressure-
sensitive adhesives. The proposed criterion is independent on small scale features of the
roughness, i.e. slopes and curvatures. According to our analysis stickiness may still depend
on the latter, but only for narrow PSD spectra. The result provides new insights on a very
debated question in the scientific community, and may serve as benchmark for the future
analysis with broad PSD spectra that are still beyond the present computational capabilities.
Acknowledgements
A.P. is thankful to the DFG (German Research Foundation) for funding the project
PA 3303/1-1. A.P. acknowledges support from ”PON Ricerca e Innovazione 2014-2020 -
Azione I.2 - D.D. n. 407, 27/02/2018, bando AIM. MC is thankful to Prof. JR Barber
and Prof. S.Abbott for some discussions. L.A. acknowledges support from ”POR Puglia
FESR-FSE 2014-2020 - Azione 1.6, bando Innonetwork. All authors acknowledge support
from the Italian Ministry of Education, University and Research (MIUR) under the program
”Departments of Excellence” (L.232/2016).
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Appendix - A
A brief outline of the Persson’s theory
We summarize the main results of Persson’s contact mechanics theory, as applied to
adhesive problem, in the variant suggested by Ref. [32]. The repulsive contact area relative
to the nominal contact area Arep (ζ) /Anom is related to the mean pressure prep (original
Persson’s theory, [37]) as
Arep (ζ)
Anom= erf
(prep
E∗√m2 (ζ)
)(23)
19
where E∗ is the elastic modulus in plane strain, and m2 (ζ) variance of profile slopes at the
magnification ζ = q1/q0, being q1 and q0 respectively the high and low wavenumber cut-off
of the surface Power Spectral Density (PSD). A better estimation of the contact area can
be obtained (see Ref. [46]) replacing in eq. (23) m2 with 〈∇h2〉/2, where for a given PSD
spectrum C (q) ⟨∇h2
⟩=
∫D(ζ)
d2q q2C (q)S (q) , (24)
being D (ζ) = {q ∈ R2| q0 ≤ |q| ≤ ζq0}, which can be interpreted as the averaged square
slope of the deformed surface (with heights h), and S = γ + (1− γ) (A/A0)2, being γ an
empirical parameter which can be taken in the range 0.4− 0.5.
For small nominal squeezing pressure or, equivalently, very large magnifications, eq. (23)
reduces to eq. (4).
At the magnification ζ, in the apparent contact area Arep (ζ) /Anom, the root mean square
roughness amplitude of the rough surface is
h2rms (ζ) =
∫q>q0ζ
d2qC (q) = 2π
∫q>q0ζ
dqqC (q) . (25)
The corresponding mean interfacial separation u (ζ) between the surfaces is related to
the pressure prep by (Ref. [47])
u (ζ) =1
2√π
∫D(ζ)
d2qqC (q)w (q)
∫ ∞prep
dp
p
×[γ + 3 (1− γ) erf2
(w (q) p
E∗
)]e−(w(q)p
E∗ )2
(26)
where
w (q) =
(1
2
∫Dq
d2q′q′2C (q′)
)−1/2
(27)
being Dq = {q∈ R2| q0 ≤ |q| ≤ q}.
In the limit of vanishing pressure, one can show that eq. (26) simplifies as (Ref. [47])
prep ≈ β (ζ)E∗exp
(− u
u0 (ζ)
)(28)
where u0 (ζ) is a characteristic length which is independent of the squeezing pressure and
20
depends on the surface roughness (and, hence, on the magnification ζ).
The distribution of interfacial separations also depends on magnification, and is obtained
by a much more complex process of integration
P (u) ' 1
A0
∫dζ [−A′ (ζ)]√
2πh2rms (ζ)
[exp
(−(u− u1 (ζ))2
2h2rms (ζ)
)
+ exp
(−(u+ u1 (ζ))2
2h2rms (ζ)
)](29)
where the apex symbol denotes differentiation by magnification.
Predictions of the above expression can be improved by substituting hrms(ζ) with (see
Ref. [32])
heffrms(ζ) =
[h−2
rms(ζ) + u−21 (ζ)
]−1/2(30)
Finally, u1 (ζ) is the (average) height separating the surfaces which appear to come into
contact when the magnification decreases from ζ to ζ−∆ζ, where ∆ζ is a small (infinitesimal)
change in the magnification, and can be calculated from the average interfacial separation
u (ζ) between the surfaces in the (apparent) contact regions observed at the magnification ζ,
and the contact area A (ζ) as
u1 (ζ) = u (ζ) + u′ (ζ)A′ (ζ) /A (ζ) (31)
Appendix - B
Comparison with Pastewka and Robbins’ theory
The Pastewka and Robbins (PR) criterion [25] is summarized in the following equation
(eqt. 10 in their paper)
h′rms∆r
2la
(h′rmsdrep
4∆r
)2/3
< π
(3
16
)2/3
' 1 (32)
where drep is a characteristic diameter of repulsive contact areas, which they estimate as
drep = 4h′rms/h′′rms, and h′rms and h′′rms are the RMS slopes and curvature. Finally, ∆r is
the attractive range which is of the order of atomic spacing (∆r ' ε for the Lennard-Jones
21
potential). When the condition given by eq. (32) is satisfied, the surfaces in mutual contact
are suggested to be ”sticky” and a finite value of the pull-off force should occur.
However, comparison with the PR results is best done considering the attractive area
(their eq. 6), reading
Aad = Arep
(16
9π
)−1/3(π∆r
h′rmsdrep
)2/3
(33)
= 2
(16
9π
)−1/3
pAnomh′rmsE
∗
(πa0
√24la/εh
′′rms
4h′2rms
)2/3
(34)
where we used their equation Arep = 2pAnom/ (h′rmsE∗) and ∆r/ε =
√24la/ε from Supple-
mentary Information of PR paper so that for la/ε = 0.05. Hence, it can be shown that,
compared to our eq. (10), their estimate corresponds to
[aV (ζ)]PR = 1.4622q0hrms
(h′′2rmsh′7rms
)1/3
h2/3rms (35)
Now for power law tail of the PSD C (q) = C0q−2(H+1), estimating h′rms =
√2m2, h
′′rms =√
8m4/3 (see Appendix - C) we obtain as for H = 0.8
[aV (ζ)]PR = 0.252ζ1/3 (36)
which results in a magnification dependence much stronger than the dependence we find (see
Fig. 2C). Hence, we conclude PR criterion and the one proposed here agree for a narrow
range of ζ (' 103), but, at very high ζ (' 106, see [48]), the difference grow since in our
theory the adhesive contact area Aad/Anom converges with ζ (eq. 10) while PR estimate
continues to grow (eq. 36).
22
Appendix - C
On random process theory
Assume the surface h (x, y) has a continuous noise spectrum in two dimensions and is
described by a Gaussian stationary process. In such case, we write
h (x, y) =∑n
Cn cos [qx,nx+ qy,ny + φn] (37)
where the wave-components qx,n and qy,n are supposed densely distributed throughout the
(qx, qy) plane. The random phases φn are uniformoly distributed in the interval [0, 2π).
The amplitudes Cn are also random variables such that in any element dqxdqy
∑n
1
2C2n = C (qx, qy) dqxdqy. (38)
The function C (qx, qy) is the Power Spectral Density (PSD) of the surface h, whose
mean-square value can been calculated as
m00 =
∫ ∫ +∞
−∞C (qx, qy) dqxdqy (39)
For isotropic roughness, using Nayak [52] definitions for the surface
mrs =
∫ ∫ ∞−∞
C [qx, qy] qrxqsydqxdqy (40)
where m00 is by definition h2rms. It can be shown by defining the PSD and the ACF (auto-
correlation function) of the partial derivatives of h with respect to x and y coordinates, and
using a relationship with the PSD of the surface, that the above spectral moments are (see
[51])
⟨(∂r+sh
∂xr∂ys
)2⟩
= m2r,2s⟨(∂r+sh
∂xr∂ys
)2⟩
= (−1)12
(r+s−r′−s′)mr+r′,s+s′ or 0,
(41)
23
depending on (s+ r − r′ − s′) is even or odd.
Nayak [52] finds for isotropic surface,
m20 = m02 = m2; m11 = m13 = m31 = 0
m00 = m0; 3m22 = m40 = m04 = m4 (42)
meaning when there is no second subscript the profile statistics for isotropic surface, which
is independent on the direction chosen.
For slopes, with the common definition of their RMS value is (also used by PR [25])
h′rms =√⟨|∇h|2
⟩=
√√√√⟨(∂h∂x
)2
+
(∂h
∂y
)2⟩
=√
2m2 (43)
where the equality depends on the result that, for an isotropic surface, the orthogonal
components ∂h∂x
and ∂h∂y
are uncorrelated.
The definition of RMS curvature h′′rms is less common, but we shall follow PR [25] in
defining
h′′rms =√⟨
(∇2h)2⟩ =
√√√√⟨(∂2h
∂x2
)2
+
(∂2h
∂y2
)2
+ 2
(∂2h
∂x2
)(∂2h
∂y2
)⟩
=√m40 +m04 + 2m22 =
√8m4/3 (44)
24
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